A linear transformation $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ satisfies $T(1,2)=(3,5)$ and $T(2,-1)=(4,-2)$.\n(a) Find the matrix $A$ of $T$ with respect to the standard basis.\n(b) Determine the eigenvalues of $A$.\n(c) State whether $A$ is diagonalizable over the real numbers.

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A = [[11/5, 2/5],[1/5, 12/5]]; Eigenvalues: 13/5 and 2; Diagonalizable over ℝ: Yes

Gce Model Questions Vol1 2024 Further Mathematics Question 15

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) satisfies \(T(1,2)=(3,5)\) and \(T(2,-1)=(4,-2)\).\n(a) Find the matrix \(A\) of \(T\) with respect to the standard basis.\n(b) Determine the eigenvalues of \(A\).\n(c) State whether \(A\) is diagonalizable over the real numbers.

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